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Symbollic Logic

by Lewis Carroll

“Book 3: Chapter 4”

Additional Information
  • Year Published: 1896
  • Language: English
  • Country of Origin: United States of America
  • Source: Carroll, L. (1896). Symbollic Logic. New York; Macmillan & Co.
  • Readability:
    • Flesch–Kincaid Level: 10.5
  • Word Count: 1,721
  • Genre: Informational
  • Keywords: math history, mathematics
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CHAPTER IV.

INTERPRETATION OF BILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS.

The Diagram is supposed to be set before us, with certain Counters placed upon it; and the problem is to find out what Proposition, or Propositions, the Counters represent.

As the process is simply the reverse of that discussed in the previous Chapter, we can avail ourselves of the results there obtained, as far as they go.

                                                      -------
   First, let us suppose that we find a Red Counter  |(1)|   |
placed in the North-West Cell.                      |---|---|
                                                     |   |   |
                                                      -------

We know that this represent each the Trio of equivalent Propositions

“Some xy exist” = “Some x are y” = “Some y are x”.

Similarly we may interpret a Red Counter, when placed in the North-East, or South-West, or South-East Cell.

                                                      -------
   Next, let us suppose that we find a Grey Counter  |( )|   |
placed in the North-West Cell.                      |---|---|
                                                     |   |   |
                                                      -------

We know that this represents each of the Trio of equivalent Propositions

“No xy exist” = “No x are y” = “No y are x”.

Similarly we may interpret a Grey Counter, when placed in the North-East, or Sount-West, or South-East Cell.

                                                        -------
   Next, let us suppose that we find a Red Counter     |  (1)  |
placed on the partition which divides the North Half. |---|---|
                                                       |   |   |
                                                        -------

We know that this represents the Proposition “Some x exist.”

Similarly we may interpret a Red Counter, when placed on the partition which divides the South, or West or East Half.

                                                        -------
   Next, let us suppose that we find two Red Counters  |(1)|(1)|
placed in the North Half, one in each Cell.           |---|---|
                                                       |   |   |
                                                        -------

We know that this represents the Double Proposition “Some x are y and some are y’”.

Similarly we may interpret two Red Counters, when placed in the South, or West, or East Half.

                                                         -------
   Next, let us suppose that we find two Grey Counters  |( )|( )|
placed in the North Half, one in each Cell.            |---|---|
                                                        |   |   |
                                                         -------

We know that this represents the Proposition “No x exist”.

Similarly we may interpret two Grey Counters, when placed in the South, or West, or East Half.

                                                          -------
   Lastly, let us suppose that we find a Red and a Grey  |(1)|( )|
Counter placed in the North Half, the Red in the        |---|---|
North-West Cell, and the Grey in the North-East Cell.   |   |   |
                                                          -------

We know that this represents the Proposition, “All x are y”.

[Note that the Half, occupied by the two Counters, settles what is to be the Subject of the Proposition, and that the Cell, occupied by the Red Counter, settles what is to be its Predi- cate.]

Similarly we may interpret a Red and a Grey counter, when placed in any one of the seven similar positions

Red in North-East, Grey in North-West;
Red in South-West, Grey in South-East;
Red in South-East, Grey in South-West;
Red in North-West, Grey in South-West;
Red in South-West, Grey in North-West;
Red in North-East, Grey in South-East;
Red in South-East, Grey in North-East,

Once more the genial friend must be appealed to, and re- quested to examine the Reader on Tables II and III, and to make him not only represent Propositions, but also interpret Diagrams when marked with Counters.

The Questions and Answers should be like this:–

Q. Represent “No x’ are y’.”
A. Grey Counter in S.E. Cell.
Q. Interpret Red Counter on E. partition.
A. “Some y’ exist.”
Q. Represent “All y’ are x.”
A. Red in N.E. Cell; Grey in S.E.
Q. Interpret Grey Counter in S.W. Cell.
A. “No x’y exist” = “No x’ are y” = “No y are x’”. &c., &c.

At first the Examinee will need to have the Board and Counters before him; but he will soon learn to dispense with these, and to answer with his eyes shut or gazing into vacancy.